Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.8% → 100.0%

Time: 4.5s

Alternatives: 7

Speedup: 1.3×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x - y}{z}, 4, -2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ (- x y) z) 4.0 -2.0))

C

double code(double x, double y, double z) {
	return fma(((x - y) / z), 4.0, -2.0);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(x - y) / z), 4.0, -2.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} + 4 \cdot \frac{x}{z} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z} + \color{blue}{\frac{4 \cdot x}{z}} \]

   3. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right) + 4 \cdot x}{z}} \]

   4. +-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left(\frac{1}{2} \cdot z + y\right)} + 4 \cdot x}{z} \]

   5. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(\frac{1}{2} \cdot z\right) + -4 \cdot y\right)} + 4 \cdot x}{z} \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right) + \left(-4 \cdot y + 4 \cdot x\right)}}{z} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z} + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-2} \cdot z + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

   9. metadata-eval N/A

      \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{\left(4 \cdot -1\right)} \cdot y + 4 \cdot x\right)}{z} \]

   10. associate-*r* N/A

       \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{4 \cdot \left(-1 \cdot y\right)} + 4 \cdot x\right)}{z} \]

   11. distribute-lft-in N/A

       \[\leadsto \frac{-2 \cdot z + \color{blue}{4 \cdot \left(-1 \cdot y + x\right)}}{z} \]

   12. +-commutative N/A

       \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x + -1 \cdot y\right)}}{z} \]

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)}}{z} \]

   14. metadata-eval N/A

       \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{1} \cdot y\right)}{z} \]

   15. *-lft-identity N/A

       \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{y}\right)}{z} \]

   16. +-commutative N/A

       \[\leadsto \frac{\color{blue}{4 \cdot \left(x - y\right) + -2 \cdot z}}{z} \]

   17. div-add N/A

       \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z} + \frac{-2 \cdot z}{z}} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{z}, 4, -2\right)} \]
6. Add Preprocessing

Alternative 2: 67.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ t_2 := \frac{4 \cdot x}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 y) z))
        (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z))
        (t_2 (/ (* 4.0 x) z)))
   (if (<= t_1 -5e+248)
     t_0
     (if (<= t_1 -1000000.0)
       t_2
       (if (<= t_1 -1.0) -2.0 (if (<= t_1 5e+159) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (4.0 * x) / z;
	double tmp;
	if (t_1 <= -5e+248) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+159) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-4.0d0) * y) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    t_2 = (4.0d0 * x) / z
    if (t_1 <= (-5d+248)) then
        tmp = t_0
    else if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 5d+159) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (4.0 * x) / z;
	double tmp;
	if (t_1 <= -5e+248) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+159) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 * y) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	t_2 = (4.0 * x) / z
	tmp = 0
	if t_1 <= -5e+248:
		tmp = t_0
	elif t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 5e+159:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * y) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	t_2 = Float64(Float64(4.0 * x) / z)
	tmp = 0.0
	if (t_1 <= -5e+248)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+159)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 * y) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	t_2 = (4.0 * x) / z;
	tmp = 0.0;
	if (t_1 <= -5e+248)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+159)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+248], t$95$0, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 5e+159], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4.9999999999999996e248 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5.00000000000000003e159

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]
   4. Step-by-step derivation

      1. lower-*.f64 64.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Applied rewrites 64.0%

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   if -4.9999999999999996e248 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6 or 5.00000000000000003e159 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]
   4. Step-by-step derivation

      1. lower-*.f64 61.1

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Applied rewrites 61.1%

      \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   if -1e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21} \lor \neg \left(t\_0 \leq 200\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -1e+21) (not (<= t_0 200.0)))
     (/ (* (- x y) 4.0) z)
     (fma (/ x z) 4.0 -2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -1e+21) || !(t_0 <= 200.0)) {
		tmp = ((x - y) * 4.0) / z;
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -1e+21) || !(t_0 <= 200.0))
		tmp = Float64(Float64(Float64(x - y) * 4.0) / z);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+21], N[Not[LessEqual[t$95$0, 200.0]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e21 or 200 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{4 \cdot \left(x - y\right)}}{z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x - y\right) \cdot 4}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - y\right) \cdot 4}}{z} \]

      3. lower--.f64 99.6

         \[\leadsto \frac{\color{blue}{\left(x - y\right)} \cdot 4}{z} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\left(x - y\right) \cdot 4}}{z} \]

   if -1e21 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 200

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} + 4 \cdot \frac{x}{z} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z} + \color{blue}{\frac{4 \cdot x}{z}} \]

      3. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right) + 4 \cdot x}{z}} \]

      4. +-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(\frac{1}{2} \cdot z + y\right)} + 4 \cdot x}{z} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(\frac{1}{2} \cdot z\right) + -4 \cdot y\right)} + 4 \cdot x}{z} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right) + \left(-4 \cdot y + 4 \cdot x\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z} + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot z + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{\left(4 \cdot -1\right)} \cdot y + 4 \cdot x\right)}{z} \]

      10. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{4 \cdot \left(-1 \cdot y\right)} + 4 \cdot x\right)}{z} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{-2 \cdot z + \color{blue}{4 \cdot \left(-1 \cdot y + x\right)}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x + -1 \cdot y\right)}}{z} \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)}}{z} \]

      14. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{1} \cdot y\right)}{z} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{y}\right)}{z} \]

      16. +-commutative N/A

          \[\leadsto \frac{\color{blue}{4 \cdot \left(x - y\right) + -2 \cdot z}}{z} \]

      17. div-add N/A

          \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z} + \frac{-2 \cdot z}{z}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{z}, 4, -2\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1 \cdot 10^{+21} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 200\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+21} \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -1e+21) (not (<= t_0 -1.0))) (/ (* -4.0 y) z) -2.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -1e+21) || !(t_0 <= -1.0)) {
		tmp = (-4.0 * y) / z;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if ((t_0 <= (-1d+21)) .or. (.not. (t_0 <= (-1.0d0)))) then
        tmp = ((-4.0d0) * y) / z
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -1e+21) || !(t_0 <= -1.0)) {
		tmp = (-4.0 * y) / z;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if (t_0 <= -1e+21) or not (t_0 <= -1.0):
		tmp = (-4.0 * y) / z
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -1e+21) || !(t_0 <= -1.0))
		tmp = Float64(Float64(-4.0 * y) / z);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if ((t_0 <= -1e+21) || ~((t_0 <= -1.0)))
		tmp = (-4.0 * y) / z;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+21], N[Not[LessEqual[t$95$0, -1.0]], $MachinePrecision]], N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e21 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]
   4. Step-by-step derivation

      1. lower-*.f64 51.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Applied rewrites 51.0%

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   if -1e21 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1 \cdot 10^{+21} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+54} \lor \neg \left(y \leq 5.7 \cdot 10^{+51}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.9e+54) (not (<= y 5.7e+51)))
   (fma (/ -4.0 z) y -2.0)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.9e+54) || !(y <= 5.7e+51)) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.9e+54) || !(y <= 5.7e+51))
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.9e+54], N[Not[LessEqual[y, 5.7e+51]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9000000000000001e54 or 5.7000000000000002e51 < y

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4 + \left(\frac{1}{2} \cdot z\right) \cdot -4}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot y} + \left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4}}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{-4}{z}} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      7. metadata-eval N/A

         \[\leadsto y \cdot \frac{\color{blue}{-4 \cdot 1}}{z} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      8. associate-*r/ N/A

         \[\leadsto y \cdot \color{blue}{\left(-4 \cdot \frac{1}{z}\right)} + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      9. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \frac{1}{z}\right) + \frac{\left(\frac{1}{2} \cdot z\right) \cdot -4}{z} \]

      10. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) + \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right)}}{z} \]

      11. associate-*r* N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) + \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z}}{z} \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) + \frac{\color{blue}{-2} \cdot z}{z} \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}\right) \cdot y} + \frac{-2 \cdot z}{z} \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-4} \cdot \frac{1}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4 \cdot 1}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-4}}{z}, y, \frac{-2 \cdot z}{z}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-4}{z}}, y, \frac{-2 \cdot z}{z}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right)} \cdot z}{z}\right) \]

      20. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right)}}{z}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \frac{\color{blue}{\left(\frac{1}{2} \cdot z\right) \cdot -4}}{z}\right) \]

      22. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{\frac{\frac{1}{2} \cdot z}{z} \cdot -4}\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-4}{z}, y, \color{blue}{-4 \cdot \frac{\frac{1}{2} \cdot z}{z}}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]

   if -1.9000000000000001e54 < y < 5.7000000000000002e51

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} + 4 \cdot \frac{x}{z} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z} + \color{blue}{\frac{4 \cdot x}{z}} \]

      3. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right) + 4 \cdot x}{z}} \]

      4. +-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(\frac{1}{2} \cdot z + y\right)} + 4 \cdot x}{z} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(\frac{1}{2} \cdot z\right) + -4 \cdot y\right)} + 4 \cdot x}{z} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right) + \left(-4 \cdot y + 4 \cdot x\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z} + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot z + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{\left(4 \cdot -1\right)} \cdot y + 4 \cdot x\right)}{z} \]

      10. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{4 \cdot \left(-1 \cdot y\right)} + 4 \cdot x\right)}{z} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{-2 \cdot z + \color{blue}{4 \cdot \left(-1 \cdot y + x\right)}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x + -1 \cdot y\right)}}{z} \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)}}{z} \]

      14. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{1} \cdot y\right)}{z} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{y}\right)}{z} \]

      16. +-commutative N/A

          \[\leadsto \frac{\color{blue}{4 \cdot \left(x - y\right) + -2 \cdot z}}{z} \]

      17. div-add N/A

          \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z} + \frac{-2 \cdot z}{z}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{z}, 4, -2\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+54} \lor \neg \left(y \leq 5.7 \cdot 10^{+51}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+126} \lor \neg \left(y \leq 4.2 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.32e+126) (not (<= y 4.2e+87)))
   (/ (* -4.0 y) z)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.32e+126) || !(y <= 4.2e+87)) {
		tmp = (-4.0 * y) / z;
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.32e+126) || !(y <= 4.2e+87))
		tmp = Float64(Float64(-4.0 * y) / z);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.32e+126], N[Not[LessEqual[y, 4.2e+87]], $MachinePrecision]], N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000002e126 or 4.2e87 < y

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]
   4. Step-by-step derivation

      1. lower-*.f64 76.4

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Applied rewrites 76.4%

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   if -1.32000000000000002e126 < y < 4.2e87

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z}} + 4 \cdot \frac{x}{z} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right)}{z} + \color{blue}{\frac{4 \cdot x}{z}} \]

      3. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left(y + \frac{1}{2} \cdot z\right) + 4 \cdot x}{z}} \]

      4. +-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left(\frac{1}{2} \cdot z + y\right)} + 4 \cdot x}{z} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \left(\frac{1}{2} \cdot z\right) + -4 \cdot y\right)} + 4 \cdot x}{z} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(\frac{1}{2} \cdot z\right) + \left(-4 \cdot y + 4 \cdot x\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot \frac{1}{2}\right) \cdot z} + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot z + \left(-4 \cdot y + 4 \cdot x\right)}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{\left(4 \cdot -1\right)} \cdot y + 4 \cdot x\right)}{z} \]

      10. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot z + \left(\color{blue}{4 \cdot \left(-1 \cdot y\right)} + 4 \cdot x\right)}{z} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{-2 \cdot z + \color{blue}{4 \cdot \left(-1 \cdot y + x\right)}}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x + -1 \cdot y\right)}}{z} \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \color{blue}{\left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)}}{z} \]

      14. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{1} \cdot y\right)}{z} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{-2 \cdot z + 4 \cdot \left(x - \color{blue}{y}\right)}{z} \]

      16. +-commutative N/A

          \[\leadsto \frac{\color{blue}{4 \cdot \left(x - y\right) + -2 \cdot z}}{z} \]

      17. div-add N/A

          \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z} + \frac{-2 \cdot z}{z}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{z}, 4, -2\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 85.8%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 4, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+126} \lor \neg \left(y \leq 4.2 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Applied rewrites 33.5%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))